→Taylor-expansion is a method of approximating a function f(x) around a point a with a polynomial of the argument x in the vicinity of a. The polynomial itself consists of the derivatives of the function of various orders.

Tn(x) = f(a) + f ‘ (a) (x-a) + f ”(a) (x-a)²/2! +  f(3)(x) (x-a)³/3! + …. + R(x). The Taylor-expansion of the exponential function is:

e<sup>x<\sup> = 1 + x + x²/2 + x³/3! + ……. + xⁿ/n!

To prove the validity of this statement consider the special case of MacLaurin-polynomials were a function is expanded around x=0. Observe the polynomials

p(x) = a +bx + cx² + dx³

p'(x) = b + 2cx + 3dx²

p”(x) = 2c + 2 *3dx.

x= 0 in each of those equalities and get expressions for the coefficients a,b,c and d. p(0) = a → a=p(0)

p'(0)  = b → b = p'(0)

p”(0) = 2c  → c = p”(0)/2

p(3) (0) = 2•3 d → d= p(3)(0)/ 2•3

Therefore the given polynomial can be written as

p(x) = p(0) + p'(0) x + p”(0)x^{2 /2+ p(3)(0)/3 !+ ……..pn(0) Xn\n!}